He was born in Elazığ in 1960. He graduated from Law School at İstanbul University. After the juridical internship, he worked as a judge at Criminal and Civil Courts in Van until 1991. He worked as a self-employed lawyer since 1992. He was the founder and manager of non-governmental organizations.

Şuay Alpay has selected as the member parliament of The Grand National Assembly of Turkey for 24th and 25th terms. During his terms, he served as a member of Constitutional and Justice Committee, President of Cameroon Friendship Group, Vice President of Azerbaijan Friendship Group, President of the Law Subcommittee responsible for Unlawful Eavesdropping. He also served at Central Decision Board of Management in Justice and Development Party he was Vice President of Local Administrations in the same period. On 30 December 2015, he has appointed as Deputy Minister of National Defense and still on the same duty.

Since June 11, 2024, he has been serving as Vice Chairman of Board of Directors of Turkish Aerospace, following the decision made at the 2023 Ordinary General Assembly meeting.

Şuay Alpay is married and father of three children.