Suppose your company has developed a lot of patents to defend itself from being sued.
Company A is sued by company B for infringement of its patents. Company B's patents are invalidated by the prior art in your company's patents. Company A also infringes your patents.
Is there any way to monetize this on this situation without suing company A?
Suing a company is what you do when you're trying to eliminate competition or as a last resort. It's much cheaper and indeed more profitable to collect royalties from company A. You'll want a lawyer involved to make sure the agreement will hold up in court, and to be available should the company decide not to negotiate with you and you're forced to sue.
IF Company A is defending itself then most likely that they know about any prior art there is out there. Follow that lawsuit - if your patents are indeed relevant prior art then most likely it will get mentioned in a court document (look for "invalidity contentions").
If that happens, go talk to a patent litigator about your options (make sure this is someone who has litigated patents).
Don't contact the other company. Too risky.