After performing a mathematical operation, for say, multiplying two integers, is it possible to access the overflow flag register in a CPU with C++ ? If not what are other fast ways to check for an overflow ?
No, generally it's impossible. Some CPUs don't even have such a flag (e.g. MIPS).
The link provided in one of the comments will give you ideas on how you can do overflow checks.
Remember that in C and C++ signed integer overflows cause undefined behavior and legally you cannot perform overflow checks after the fact. You either need to use unsigned arithmetic or do the checks before arithmetic operations.