Can you deposit a large check into your spouse's account that does not have your name on it?

Yes. There are no restrictions as to who can deposit cash into a checking account. Anyone can deposit cash into anyone else's account as long as they know the account number of the account into which they want to deposit cash. For ex: I can walk into the bank branch where you hold a checking account and deposit $500 into it if I know the account number. Nobody will ask or question me as to why I am doing so…

Since in your case it is only your spouses' account no one is going to ask you anything. But, since the amount is high (as you say) she may have to show proof of income if the bank asks her to do so.