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