No, movement speed has absolutely no relation to attack or damage.
Considering that there are ways to boost speed to stratospheric levels (literally, in the sense that you can exceed Earth’s escape velocity; before an erratum, you could even exceed c), that’s probably for the best.
Instead, 3.5 and Pathfinder model the extra oomph you can put behind your attacks when moving at high speed via the Charging combat maneuver. Charging gives you a +2 to attack (and a −2 to AC), but you get the same benefit when you charge no matter if your speed is 15 ft. or 150 ft. And you don’t get the benefit without charging, even if you move farther in a single move action than someone else might while charging.
The only innate effect of high movement speed, aside from the ability to move further in a round, is the (rather large) bonus you receive to Jump checks.
Specific feats and class features may reference movement, however. For instance, the Scout (Complete Adventurer) has a Skirmish class feature, which deals extra damage as long as you move 10 feet, and can take the Improved Skirmish feat for additional bonus damage if you move 20 feet. The Expeditious Dodge feat (Races of the Wild) gives you an AC bonus as long as you move 40 feet.
I haven’t found any options yet that scale with speed, but as I pointed out, speed also increases Jump checks. The Roof-Jumper feat gives you +1d6 damage per 10 feet you fell from above the target. If you have a sky-high Jump check, then you can jump, well, into the sky – and stack extra d6’s on your attack. Note that you’re not allowed to slow your fall (e.g. feather fall) if you want the damage, though, which means you may take falling damage (Tumble, as well as certain items, can mitigate falling damage, however). All that said, however, each 10 feet of vertical jump height requires a DC 40 higher, and each +40 bonus to Jump from movement requires 100 ft. more movement speed. You can get d6’s of damage much more easily than you can get 100s of feet of movement speed...
See Are there rules for dropping on an enemy as an attack? for more ways to get extra damage from jumping (though only Roof-Jumper directly scales with height).