If you owned the home (or co-owned the home) for at least 2 years, and lived in the home at least 2 of the past 5 years, you qualify for the exclusion. Your problem is determining your basis, and how much the gain actually is.
If the total sales price was $250,000 or less, then even if we assume there is zero basis, your entire gain is excludable and we don't really have to consider the basis at all.
However, if the sales price was more than $250,000, we need to determine your basis, so we can figure out what your gain is. If your gain is more than $250,000, you pay tax on the difference. (For example, if your basis is $50,000 and the sales price is $350,000, you have a $300,000 gain. You can exclude $250,000 and pay capital gains tax on the rest.)
To start with, your mother's basis was the fair market value on the day your grandmother died. You can get an appraisal or market analysis from a real estate professional based on historical records. So that's the lowest your basis might be. The basis is increased by the cost of any permanent improvements your mother made after she was the owner.
Then, because this was a quitclaim deed instead of a life estate, I think your basis will be your mother's basis as of the date of the gift. Then you can add the cost of any permanent improvements you made. Your gain is the difference between your sales proceeds and your basis.
