As an abstract spherical-chickens-in-a-vacuum-type question, then no, basic relativity says not.
But that isn't much fun. ..
First up, what do you mean by speed? If you mean speed along surface of earth then you have a chance. Since the earth is curved you are always accelerating, and you could measure the drop in g as you speed up. So long as you don't go fast enough to go into orbit... Of course the is some degeneracy between speed and altitude for this.
Of course. If we are moving on a true straight line it is easier still as you will move through the earth's gravity as you pull away from the surface. Or burrow beneath it!
A more practical method (that is actually used) would be dead reckoning. If we assume (or assert!) that the vehicle was stationary when you got in then you can in principle detect every acceleration the vehicle makes from then on. Integrating these will give you your current velocity. Of course the accuracy of this method decreases with time as the errors simply accumulate.
Finally, if our vehicle is ground based, then an oddity of fiction could at least tell us if we are probably moving or stationary. Friction between surfaces is typically very different for static and dynamic friction, with static being stronger. This means that when something slides to a stop (be it a sled or break discs) it almost always jolts slightly at then end as the stronger static fiction kicks in. You will feel this on buses and trains frequently. Although careful control can minimise the jolt, looking out for it with a sensitive accelerometer would give you a strong hint you had stopped.