I just started my first foray into the world of self-employment after moving overseas for my husband's military orders. I was employed W2 by the company I work for until the end of April; now I'm a 1099 independent contractor for the same company. I started work as an IC on May 1st and will be sending my first invoice on June 1st, so I won't be getting paid until June.
Am I required to make an estimated tax payment on June 15 for the work performed in May, even if I don't get paid until June?
When you receive your first payment is not really factor.
What you need to consider is what your total taxes will be for 2019. This will include income tax AND self-employment tax. The SE tax is something you haven't had to deal with before. It is roughly just under 15% of your net income from self-employment.
You need to estimate your total income tax plus SE tax. From that, you would subtract any withholding (yours and your husbands), less any credits, pre-payment credits, etc. Then you would spread out the remaining tax over the estimated payments. It might be easier to include some payment in the June filing, that way the remainder of the payments won't have to be as large. 2019 won't be as large as what probably 2020 will be. You have your W2 withholding of Jan - April to help pay the taxes...but, the SE tax will greatly increase your tax bill.
See the following link for some more information. The link also has sub-links for forms and further information