No, cutting or trimming nails — both fingernails and toenails — is prohibited while in the state of احرام. This is one of the established prohibitions (mahdhurat) of احرام, 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 حاجی may choose among these three options.
To avoid any issues, scholars strongly recommend trimming nails before entering the state of احرام. As part of your pre-احرام preparation (along with bathing, grooming, and applying perfume to the body but not the احرام garments), trim all fingernails and toenails. This is actually from the Sunnah of preparing for احرام. If your حج or عمرہ lasts several days and nails grow uncomfortably long, you must wait until you exit the state of احرام to trim them.