Strictly defined, a hysterectomy is removal of only the uterus. But sometimes the ovaries are also removed, depending on the reason for the surgery and age of the woman.
If the ovaries are also removed, there would no ovarian or uterine cycling, true. But without the negative feedback effect of ovarian steroids (progesterone and estrogen), levels of both FSH and LH would be persistently, abnormally high. The same happens in postmenopausal women (ovaries still present but no longer producing steroids) and spayed pets.
If the uterus is removed but the ovaries left in, the ovaries would continue to mature follicles that release estrogen, ovulate, luteinize and produce progesterone, etc, etc. Even though there is no uterus for the steroids to act on (and no menstruation), the steroids would feed back to the anterior pituitary gland and FSH/LH levels would fluctuate in the normal patterns.
Without the obvious sign of menstruation, it would be tricky for the woman still with ovaries to know whether or not she had reached menopause. The first woman (ovaries removed) would be thrown into a 'surgical menopause', an abrupt (and nasty) withdrawal of all ovarian steroids.