Veteran Families Health Services Act of 2021 This bill addresses fertility treatments, counseling, and other health-related programs for veterans and members of the Armed Forces. Specifically, the bill requires the Department of Defense (DOD) and the Department of Veterans Affairs (VA) to provide fertility treatments and counseling to covered veterans and active duty service members or a spouse, partner, or gestational surrogate of such veteran or service member. A covered veteran or service member is a former or current member of the Armed Forces who has an infertility condition, unless it is demonstrable that the individual was completely infertile prior to serving in the Armed Forces. In the case of in vitro fertilization treatment, DOD and the VA may not furnish more than three completed cycles or six attempted cycles, whichever occurs first. DOD must establish procedures related to gamete retrieval and storage. Additionally, DOD must provide service members with the opportunity to cryopreserve and store their gametes prior to deployment to a combat zone or a hazardous duty assignment. DOD and the VA must share best practices and facilitate referrals for the furnishing of fertility treatment and counseling to eligible individuals. Fertility counseling and treatment are included within authorized VA medical services. The VA may pay the adoption expenses (for up to three adoptions) for a covered member who is enrolled in the VA health care system. The VA must facilitate specified reproduction and infertility research conducted collaboratively by DOD and the Department of Health and Human Services.