No, cutting or trimming nails — both fingernails and toenails — is prohibited while in the state of ইহরাম. This is one of the established prohibitions (mahdhurat) of ইহরাম, agreed upon by all four schools of jurisprudence. The prohibition applies equally to men and women and covers removing nails by any method, not just cutting.
If a nail breaks on its own or is hanging painfully, scholars generally permit removing the broken part without penalty, as this falls under genuine necessity. However, deliberately cutting even a single nail requires fidyah (penalty): fasting 3 days, feeding 6 poor people, or sacrificing a sheep — the হাজী may choose among these three options.
To avoid any issues, scholars strongly recommend trimming nails before entering the state of ইহরাম. As part of your pre-ইহরাম preparation (along with bathing, grooming, and applying perfume to the body but not the ইহরাম garments), trim all fingernails and toenails. This is actually from the Sunnah of preparing for ইহরাম. If your হজ্জ or উমরাহ lasts several days and nails grow uncomfortably long, you must wait until you exit the state of ইহরাম to trim them.