Unless you are putting conditions on the hiring, I see no ethical need to inform the institution. There is no conflict of interest in the hiring process, though it might arise later.
If you won't accept an offer unless both are accepted or if you won't accept an offer unless both are placed in the same group, then you should inform them. But if you are flexible about it then you are just two people applying for positions.
Working together isn't a conflict unless there is some competition involved.
If you are both accepted to the program then you can both apply to the group, at which point disclosure would seem necessary.
This is the classic two-body problem. It is sometimes helpful, however, to disclose a relationship when it is necessary for both to be "hired". Useful because if one of you is very highly desired then the other's qualifications might not be quite the same issue as otherwise. That is a judgement call for you and ultimately a judgement call for them.
Note that I've assumed that applying to the program and applying to the group are distinct. If it is the same, then earlier disclosure might be required and whether it works for or against you depends on a lot of judgements. But if you won't accept an offer from one but not the other, then you should say so to avoid forcing people to waste time unproductively in the process. In a small group, I'd guess it works against you unless the group is expanding (or willing to expand).
And, advice not asked for: the two-body problem can often be solved by applying to places that have a lot of nearby universities. That often means large metropolitan areas, of course, which can be expensive or involve a lot of commuting effort. And, working at nearby universities doesn't foreclose the possibility of working together, necessarily, though that is more true for faculty than doctoral students.