Playing microtones on a concert flute is as simple as rotating the head plate towards or away from your lips (not twisting against the body!) to decrease or increase the effective length. Increasing the distance sharpens the note, and vice versa. This requires extremely-minute precision and fine muscle memory for instant execution. You may also need to change your embouchure to avoid whisper tones.
For example, suppose you want to play a tone of 449 hertz (A4+35¢). The closest note in the 12 tone tempered scale is A4 at 440 hertz, so you would need to sharpen this note by approximately a sixth-tone, or 35 cents. You would accomplish this by rotating the head plate away from your lips. This technically increases the effective length of the flute, which physics dictates would normally flatten a note, though contrary to common sense it does the opposite.
Additionally, some flutes come with holes in the keys ("ring keys") to facilitate quarter tone use. However, I do not recommend you purchase this type of flute unless if you have a desire to play in 24 EDO. These holes allegedly change the tone-color, which might be considered an undesirable side effect.
Above: a hand-made PVC flute tuned to 7 EDO, which can also play many intermediate notes of 14 EDO with cross-fingerings.
The flute was completed in under 12 hours and required under 1 USD of raw materials. One could feasibly hope to make multitudes of these inexpensive flutes for every EDO or other tuning scheme.
Eva Kingma and the quarter-tone flute - video demonstration of a modified flute able to play quartertones