Can I make a latch using 2 AND gates, instead of NAND gates?
The simple answer is that you can't make a latch from just AND gates because it won't work. You can't do it with just OR gates either.

Now, demonstrating that it doesn't work is a great homework question and I don't want to rob you of that learning experience. Make a truth table and figure out how the next output values will depend on the input values and the previous output values.