What do you think of this argument?
I think it is pretty easily dismantled by simple logic, not theology. Since logic is in the purview of philosophy, you're quite on-topic here.
The fallacy of your argument is that you have not only one assumption, but that you have two:
- Assumption A: "God does not exist."
- Assumption B: "Something that is impossible cannot be conceived of."
From these two assumptions, you have deduced a contradiction. This allows you to say that
(A and B) is false, or equivalently, that
(A is false) or (B is false) or (both A as well as B are false) to spell it out as verbatim as possible.
(A and B) is false you can not deduce that
A is false; but this is what your argument is erroneously doing.
So to make your argument hold tight, you have to prove that your assumption "Something that is impossible cannot be conceived of by the mind." is in fact true. It is not enough to say that your mind cannot do it, you must prove that every mind cannot do it.
(Note that there may be further errors, even if you manage to solve this first one.)
N.B.: As mentioned in the comments, part of the problem with the assumption is that it is unclear what "impossible" means. If OP wants to make the argument watertight, then this needs to be defined (and then the assumption may or may not be self-evident; and the rest of the argument will need to be checked again to verify that the logical conclusions follow from it, and so on). The TLDR of this answer is that it's too complicated/complex to simply say "it's true".