The Waltz is danced to music with 3 beats to 1 measure (3/4) and most figures are danced 1 step to 1 beat. Counting is 123 123 123 etc., but some figures use divided beats, the simplest being the whisk and chasse, where the count would be 123 12&3. Several figures are danced over 2 measures, such as Natural Turns & Reverse Turns. Each has 6 steps, but each of 1 beat, and the musical count would still be 123 123, not 123456.
Foxtrot is danced to music with 4 beats to 1 measure (4/4), but steps are counted as slows or quicks. A quick is 1 beat, but a slow is 2. The basic figures of foxtrot are of different lengths: Feather Step & Three Step, both 1 measure of 4 beats, counted SQQ; Reverse Turn 2 measures, counted SQQSQQ and Reverse Turn 2 1/2 measures, counted SQQSSS (all as Guy Howard).
I'm not sure that I can see how either dance can be considered to have 6 beats to 1 measure. Neither do. The only dance that can follow that pattern is Viennese Waltz, using music in 6/8 metre (although 3/4 at a faster tempo is more common). The 6/8 time signature is in compound time, with a quaver (eighth note) pulse, so that the six beats fall into two accented groups: 123 456 (which is why the VW fits), and not just like 3/4, with its crotchet (quarter note) pulse. If 3/4 was counted in quavers, you would get 1& 2& 3&, rather than 6/8's 123 456.