rent

If security deposit is refundable, then it is not income to you. You do not report it on your tax return until and if it becomes non-refundable. 

On your 2020 tax return you will need to report only that income that you received in 2020. Sounds like rental income received in November and December. 

Here are a couple of Help Articles that have additional information:

Where do I enter income and expenses from a rental property?

Shat kinds of rental property expenses can I deduct?

I'd like to throw in a few tid-bits if I may, since I get the impression that you are a first time landlord.

Run (don't walk) to Barnes & Nobel and purchase a book title "Landlord/Tenant Rights". Another name for the same exact book is "Landlord's Rights & Duties in [state]" where [state] is your specific state where the property is located. The one I have is "landlords' RIghts & Duties in Florda, 10th Edition". You can also find this book on Amazon, and it's also available for purchase on Kindle. Understand I'm not recommending this book as a convenience, but as a requirement to protect your interest.  It will be the cheapest investment, as well as the most important investment you will ever make in rental property.

i started renting the end of Oct the tenant paid me Security deposit

Understand that the security deposit is not your money, and it never will be until you make legal claim to it. Therefore, it does not get included or counted as income to you, until such time you make legal claim to it. In my state, I am required to keep the security deposit separate from other funds. Therefore I opened a separate escrow account and deposited the funds in that account. In my case, the escrow account does not pay any interest. But if it did, that interest belongs to the tenant. I could provide more details on this, but you'll have to check the laws of your state to see how to deal with it so that you're not violating any laws, rules or ordinances on the security deposit issue.

I did not touch that money and only took what was owned me on an monthly basis.

That's irrelevant. Rental income is reported in the tax year it is actually received. It does not matter what tax year that rent payment may be *for*.  Once the renter has paid that money they have relinquished their control of that money, and you now have obtained control of that money. Once you have control of it, it's reportable income for you.  So even if the tenant prepaid you 3 years of rent in 2020, you have to claim every penny of that income on your 2020 tax return.

Now on the tax front, getting that tax return absolute perfect on the SCH E in your first year of dealing with rental property, perfection is not an option. It's an absolute must. Even the tiniest of mistakes can grow exponentially over time. Then when you catch the error years down the road, the cost of fixing it will be expensive. Generally, errors are not realized until filing the tax return for the tax year the property was sold. So if you made a tiny mistake on the initial first year return for that rental and then sell the property 10 or more years later, chances are that error will have grown exponentially and the cost of fixing it is going to be quite painful on the financial front.

Now I'm not trying to scare you here. If you use the program the way it's designed and intended to be used, you should not have any problem really. But I can't stress enough the improtance of reading the small print on each and every single screen in the Rental & Royalty Income (SCH E) section of the program, so that you make the proper selections and enter the correct data. Of course, if you are unsure of anything (and I mean *ANYTHING*) then by all means please ask. Meanwhile, I've included some information below that will help clarify things for you a bit better than the program does in my opinion. (and we all know what opinions are like!)

Rental Property Dates & Numbers That Matter.

Date of Conversion - If this was your primary residence or 2^nd home before, then this date is the day AFTER you moved out, or the date you decided to lease the property – whichever is later.
In Service Date - This is the date a renter "could" have moved in. Usually, this date is the day you put the FOR RENT sign in the front yard.
Number of days Rented - the day count for this starts from the first day a renter was contracted to move in, and/or "could" have moved in. That would be your "in service" date or after if you were asked for that. Vacant periods between renters do not count for actual days rented. Please see IRS Publication927 page 17 at https://www.irs.gov/pub/irs-pdf/p527.pdf#en_US_2020_publink1000219175 Read the “Example” in the third column.
Days of Personal Use - This number will be a big fat ZERO. Read the screen. It's asking for the number of days you lived in the property AFTER you converted it to a rental. I seriously doubt (though it is possible) that you lived in the house (or space, if renting a part of your home) as your primary residence, 2nd home, or any other personal use reasons after you converted it to a rental.
Business Use Percentage. 100%. I'll put that in words so there's no doubt I didn't make a typo here. One Hundred Percent. After you converted this property or space to rental use, it was one hundred percent business use. What you used it for prior to the date of conversion doesn't count.

RENTAL PROPERTY ASSETS, MAINTENANCE/CLEANING/REPAIRS DEFINED

Property Improvement.

Property improvements are expenses you incur that Improve, restore, or otherwise “better” the property. Basically, they retain or add value to the property.

Betterments:
Expenses that may result in a betterment to your property include expenses for fixing a pre-existing defect or condition, enlarging or expanding your property, or increasing the capacity, strength, or quality of your property. An example of a pre-existing condition or defect in this context would be something such as foundation repair (slab jacking) or some other, hidden and costly, anomaly.
Restoration:
Expenses that may be for restoration include expenses for replacing a substantial structural part of your property, repairing damage to your property after you properly adjusted the basis of your property as a result of a casualty loss, or rebuilding your property to a like-new condition.
Adaptation:
Expenses that may be for adaptation include expenses for altering your property to a use that isn’t consistent with the intended ordinary use of your property when you began renting the property. Adding a wheelchair ramp would be an example.

Expenses for these types of costs are entered in the Assets/Depreciation section and depreciated over time. Property improvements can be done at any time after your initial purchase of the property. It does not matter if it was your residence or a rental at the time of the improvement. It still adds value to the property.

To be classified as a property improvement, two criteria need to be met:

1) The improvement must become "a material part of" the property. For example, remodeling the bathroom, new cabinets or appliances in the kitchen. New carpet. Replacing that old Central Air unit.

2) The improvement must retain or add "real" value to the property. In other words, when the property is appraised by a qualified, certified, licensed property appraiser, he will appraise it at a higher value, than he would have without the improvements.

There are rules that allow you to just flat-out expense and deduct some property improvements instead of capitalizing and depreciating them, if the total cost of the improvement was less than $2,500. It’s referred to as “safe harbor di-minimis” But depending on the specific situation, this may or may not be beneficial. Just be aware that not every property improvement that cost less than $2,500 qualifies for this. If this interest you, the rules can get complex. So a good place to start reading is on the IRS website at https://www.irs.gov/businesses/small-businesses-self-employed/tangible-property-final-regulations. The stuff on di-minimis starts about one page down.

Cleaning & Maintenance

Those expenses incurred to maintain the rental property and it's assets in the useable condition the property and/or asset was designed and intended for. Routine cleaning and maintenance expenses are only deductible if they are incurred while the property is classified as a rental. Cleaning and maintenance expenses incurred in the process of preparing the property for rent are not deductible.

Repair

Those expenses incurred to return the property or it's assets to the same useable condition they were in, prior to the event that caused the property or asset to be unusable. Repair expenses incurred are only deductible if incurred while the property is classified as a rental. Repair costs incurred in the process of preparing the property for rent are not deductible.

Additional clarifications: Painting a room does not qualify as a property improvement. While the paint does become “a material part of” the property, from the perspective of a property appraiser, it doesn’t add “real value” to the property.

However, when you do something like convert the garage into a 3^rd bedroom for example, making a  2 bedroom house into a 3 bedroom house adds “real value”. Of course, when you convert the garage to a bedroom, you’re going to paint it. But you will include the cost of painting as a part of the property improvement – not an expense separate from it.