In my biochemistry textbook the terms alpha helix and alpha subunit are both used. Are these two terms synonymous? For instance, hemoglobin has an alpha and beta subunit, are these the same thing as alpha helix and beta pleated sheet?
These terms are completely unrelated. Alpha-subunit is an arbitrary name for a protein subunit in a multi-subunit protein complex (one having a quaternary structure like haemoglobin or G-protein etc).
Alpha-helix is a type of secondary structure.
Addition by David:
For those who may be unfamiliar with this, alpha, beta, gamma etc. are Greek letters of the alphabet (equivalent to English a, b, c) — also written α, β, γ. They are sometimes used in science to enumerate or differentiate between a series of related things — proteins, small molecules etc.