If you've bought a queen-size mattress, but you have a king size bed frame, you don't necessarily have to buy a new bed frame. A king bed frame holds a mattress that is 76 inches wide and 80 inches long. Your queen mattress is only 60 inches wide, but it is also 80 inches long. The frame will hold your bed, since it's bigger, but some of the frame will be sticking out. The key to making this work is to make it look like it's intentional, by turning it into a platform bed.
Measure the depth of the bed frame. Your bed frame likely consists of a metal frame that your mattress sits in. Around the edge, there should be about 1 to 2 inches of metal that will cover your box spring. Measure this.
Purchase planks of wood to fill the bed frame. People will see the pieces on the sides, so these should be solid, but you can piece together the parts in the middle that the mattress covers. Wood should be as thick as the measurement that you took in step one and 80 inches long. You can add multiple pieces together to fill the frame.
Center the queen mattress on the frame. You should have 8 inches of wood on either side of the mattress. You do not need the box spring.