Any way to treat a cavity under the gums without cutting the gums?

Maybe: It may be able to be fixed with a crown. You may need to have the gum cut and a procedure called crown lengthening. This will give the restorative dentist enough tooth structure to do a restoration and not encroach into the gum attachment.

Cavity under gums: It depends how far under the gums and how big the cavity is... The dentist may be able to push the gums aside temporarily.

Most probably not. : It mainly is dependent upon location of the cavityy, extent/severity, and the type of the treatment /restoration you need to have on such teeth. (the treatment flap surgery, or crown lengthening, or gingivectomy which are mainly the possible treatments for such cases are not as bad as what you think!) and they may even save you from future hassles when you do these treatments once and for all.

Possibly: Depending on how far below the gumline the cavity is, your dentist can sometimes move some of the gum tissue away without cutting it to repair the cavity. However, if the cavity is deep below the gumline, the only way to access and take care of it will be to move the gums out of the way. Speak with your dentist about your particular situation to see what can be done.