Your terminologies and your conditional question/statement are a bit confusing.

I suppose your "first residence" vs "second residence" actually mean "former home" and "current home".

You added that the taxpayer "Would like to move to Maine before two year" but didn't your client already move in July 2018?  I suppose you meant that the plan is to stay in MA for two years after the move but your client may end up staying for a shorter period - the key question is what is the primary reason for selling the current home (see below).

It is also not clear what your question "Move for job qualify if actually got the job in 2018 and sell before Aug, 2020" actually means.  My understanding is that the taxpayer sold the former residence in July 2018 due to a change of employment and claimed the exclusion.  The reason for selling that former residence is no longer relevant now, just that the timing of when the last exclusion was claimed must still be considered.

To claim a partial exclusion under §121(c) and §1.121-3, your client's primary reason for selling the current home must meet the conditions prescribed in the regs.  You did not, however, explain in your question what the reason is or your basis for claiming a partial exclusion.