In short , answer is "No".
If you look into the "S1-AP based handover required" message, there is an IE "target Id". Based on the Target Id , source MME get to know whether it has S1-connection with target eNodeB or not. It may have S1-connection with target eNodeB or the MME may not aware about the target eNodeB . In case of S1-connection availability, MME triggers "Handover Request" towards target eNodeB . As result, there is no change in MME.
Now consider the another scenario, in which source MME doesn't know about the target eNodeb. MME creates S10 interface based forward relocation request (having target Id information ) and sends to target MME. This is the scenario where MME gets changed.