Unfortunately no, but check with your GM
In principle, Inspiration is not limited to investigative ability checks:
An investigator typically uses these powers to aid in their investigations, but can also use these flashes of inspiration in other situations.
So it should be possible to use it on such a check. The reason why this doesn't work is rather technical, due to the way combat is organized. Using inspiration requires a free action:
As a free action, he can expend one use of inspiration from his pool to add 1d6 to the result of that check, including any on which he takes 10 or 20.
And therein lies a problem: normally you can only take actions on your turn. From the Combat rules:
Each round’s activity begins with the character with the highest initiative result and then proceeds in order. When a character’s turn comes up in the initiative sequence, that character performs his entire round’s worth of actions. (For exceptions, see Attacks of Opportunity and Special Initiative Actions.)
So, first you establish initiative with the initiative roll, and only then, on your turn, you can take actions. Exceptions to this are Attacks of Opportunity and the Special Initiative Actions Ready and Delay. Using inspiration is neither of those. (And it is questionable if you even could Ready an action outside of combat).
Because you first have to establish initiative before you can take your actions, you cannot take this free action until you have established initiative, and at that point it is too late — you already have established initiative. The technical sequence of steps of combat is getting in your way here.
That said, free actions consume a very small amount of time and effort and for some, such as speaking a few words, the rules allow you to take them even when it isn't your turn. So check with your GM, if they allow you to use Inspiration when it isn't your turn, too. If they do, then it would work.