Hello

No. After modal verbs - here could - you need an infinitive. In this case it's a past event so you need the perfect infinitive : have+ past participle.

So the correct sentence would be : We were wondering what could have possibly gone wrong or We were wondering what could possibly have gone wrong - I prefer the second, but both are acceptable.

thank you!! I was even wondering at the possible use of had + past participle. but of course that doesn't exist