Is there a way to calculate the four 3-month LIBOR rates (spot and forward) given a 1-year spot LIBOR rate and a 1y1y forward LIBOR rate?
You can calculate the risk-free 2-year LIBOR rate based on what you have. You can impute the 3-month LIBOR rate if you have an overnight rate to start with. Then the 3-month LIBOR rate would depend on your assumptions (linear, constant forward, etc)
For the 3-month forward LIBOR (forward starting point within 21 months), it's gonna be the result of imputation as well.
You are basically trying to generate a curve based on two points.
From 1 year interest rates you can calculate financial equivalent interest rates for other maturities, but not the interest rate of those maturities.