No. 1080p60 would be a 60fps 1080p. 1080p30 would be a 30fps 1080p. 1080p is 1,080 lines of resolution (which depends on the number of pixels in the camera) progressive scan (as opposed to 1080i, which is interlaced scan - 1080i60 is 60 half frames, or fields, per second). The number after the letter is the frame rate.
The frame rate doesn't determine the definition of the picture. Look at old movies played at 18fps. People look as if they're moving in jerks, not smoothly. Yet the film is MUCH higher than 1080 (vertical lines, or vertical silver particles in the case of film). And the resolution of 4,200fps high speed photography is no greater than you can get with an old brownie camera. It's like mixing mpg and mph. Resoultion isn't frame rate, 1080 is actually 1080X1920 It's the number of lines (vertical) and dots (horizontal) of resolution, and is directly related to the sensor in the camera.