Typically yes you would pay interest and principal payment the month of closing.
This could of course depend on the terms of your loan.
You can typically call your lender...tell them what day you will close and they can give
you payoff amount.
Remember how you got that "free" month when you got the loan. Well it's not really free.
Typically loans are set up where you have the money and pay the interest for use of the money.
So if you typically pay 1st of month, that is last month's interest.
If you close on 25th of the month, then you owe the outstanding loan balance AND 25 days of interest
for which you had use of the money.
Hope that helps.