No, the only way to know the number of eggs a woman has reliably is through the blood test discussed in this article, or by observing the antral follicles by ultrasound.
There is a home test to measure basal temperature, but it is only useful to know if ovulation is occurring.
Read the full article on: Can a Woman’s Egg Count Be Measured with a Blood Test? ( 59).

Zaira Salvador
B.Sc., M.Sc.
Bachelor's Degree in Biotechnology from the Technical University of Valencia (UPV). Biotechnology Degree from the National University of Ireland en Galway (NUIG) and embryologist specializing in Assisted Reproduction, with a Master's Degree in Biotechnology of Human Reproduction from the University of Valencia (UV) and the Valencian Infertility Institute (IVI)
License: 3185-CV