To expedite or accelerate one's pace are effective synonyms for quicken one's speed. The phrase quicken one's pace means to increase the speed at which someone does something. Synonyms for this phrase would include: hasten, hurry, rush, and speed up. Another means of quickening one's speed would be to pick up the pace. This phrase is frequently used in the context of athletic competition or quickening the tempo of music. To quicken one's steps can also be interchangeable with quickening one's speed, as it implies taking larger strides or moving quicker with one's feet. Find a synonym that works best for what you're trying to communicate, and add variety to your writing that can keep a reader engaged.