No, cutting or trimming nails — both fingernails and toenails — is prohibited while in the state of الإحرام. This is one of the أُسس prohibitions (mahdhurat) of الإحرام, 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 فدية (penalty): fasting 3 days, feeding 6 poor people, or sacrificing a sheep — the الحاج may choose among these three options.
To avoid any issues, scholars strongly recommend trimming nails before entering the state of الإحرام. As part of your pre-الإحرام preparation (along with bathing, grooming, and applying perfume to the body but not the الإحرام garments), trim all fingernails and toenails. This is actually from the السنة of preparing for الإحرام. If your الحج or العمرة lasts several days and nails grow uncomfortably long, you must wait until you exit the state of الإحرام to trim them.