it is decidable, you can prove like this,
1-make PDA for palindromic strings and find corresponding CFG LC
2-find language for given DFA, say it LR
3-find Lfinal = LR intersection LC, which will also be CFG
4-check if Lfinal is empty
5- if empty your dfa does not accept any palindrome else your dfa accept some palindrome
steps 1 to 5 are all decidable because we have algorithm for each steps, so finally this qsn " M is a DFA that accepts some palindrome ?" is decidable