Rik, here is my best shot.

Following the Fibonacci sequence (1, 1, 2, 3, 5 ,8, 13 21, 34, ...), I created a vertical line of unit length (here 50px).
I copied that and turned it 90° (horizontal) and created new lines the length matching the Fibonacci Sequence (above).

After running out of paper (A2), I switched to the Shape Tool and with Smooth Join, I touched each of the corners, moving along the line from in to out.

The trick is to choose a source point, just off the start point and to finish beyond the end point with a nearby sink point; these allow the sequence to be effectively truncated.

Fibonacci Curve.xar.

No guarantees it is right as the curves are Bezier but possibly more accurate than quarter circles.

Acorn