No. They can bring it up in opening. Then they can try to put on witnesses to attest to her "bizarre" or "unstable" behavior. The judge will decide whether it's relevant testimony or to allow it in.
In these cases we have sometimes seen a lot of latitude given to the defense though. Think about the Jodi arias trial. (And I know she testified but I'm talking in terms of what horrible, unfounded theories they allowed to be explored to dirty the name of the deceased).
They can try to put on "experts" to try to say that video or audio footage or SM posts or texts, voicemails, emails, whatever, evidence abusive behavior. They did that in the arias case.
In order to really prove she had mental health issues that could've led to her going off the deep end they'd likely though, have to have her doctors up there and/or evidence of calls to 911 due to suicidal ideation, threats, etc., attempts at psych holds, etc. Things we've seen in cases where women have annihilated their families.
And I have yet to see one such case where that kind of thing wasn't known and reported on within DAYS of the homicides. Not one.
If they go this route it's going to be a tough row to hoe. Because there's going to be a ton of evidence and testimony that evidences she was stable, IMO. Her close friends like NUA, who saw and spoke with her daily. The Thayers. School staff. Fellow parents. Colleagues in the business. Her doctors. Just the fact that her husband left the kids alone with her regularly.
Her life and how those around her interacted with and reacted to her evidence stability.
There's going to have to be a lot more than, "yeah she was successfully treated for some depression and anxiety issues. She continued with treatment and reported no significant problems."
And as I've said, I've yet to see a female family annihilator case when that stuff didn't come out very quickly.
Impressions or gossip from those desperate to believe CW won't be enough.