Several studies have asked this, with differing results. But some find that testosterone does appear to vary through the year to a moderate degree. It usually peaks around the Autumn, then drops through the winter to a low in the Spring.
Paris 1975: five healthy medical students and biochemists living in Paris age 26-32. “The peak of plasma testosterone was in November (909 ± 42.7 ng/dl) and the trough was in May (640 ±51.1 ng/dl)”) (source)
1974-1975 in Nijmegen, The Netherlands: 15 healthy men age 24-45. Testosterone averaged 666 ng/dl in April 1974, 744 in July, 762 in October, 665 in January 1975, and 631 in April 1975. (source)
That was the average, but there was a lot of individual variation. Some men peaked in July and some peaked in October. One man looked like this: April: 781 ng/dl, July: 1084, October: 1144, January: 586, April 778. That guy didn’t like January much! However, most of them did not change as much as that. Maybe that guy had very low Vitamin D or something.
Boston, Massachusets in the USA in the 2000’s – Among 30-79 year old men, testosterone rose until July, reaching 470 ng/dl, and then began dropping to a low in December of 385 ng/dl (source) The study didn’t find the differences to be significant however.