Is it possible to get a braid representation for a general Montesinos link with small number of strands? I know by Alexander's theorem it is possible to braid any link but is it possible to find a braid of index three for a Montesinos link?
No, the braid index of a Montesinos link can be as large as one wishes.
One way to see this is as follows.
There is a lower bound for the braid index in terms of the Homfly polynomial due to Morton and FranksWilliams. In terms of the Jones polynomial it says $$b\ge\frac{d}{2}+1,$$ where $b$ is the braid index and $d$ the degree of the Jones polynomial (i.e., the difference between largest and smallest exponent).
On the other hand, Stoimenow has computed the Jones polynomial of Montesinos links in https://projecteuclid.org/download/pdf_1/euclid.jmsj/1191591855 and from his computations one sees that the degree $d$ can be as large as one wishes.

4$\begingroup$ another way of seeing this is the following: A lower bound for the braid index is the bridge number (this is obvious) and the bridge number of Montesinos knots was computed by Boileau and Zieschang. $\endgroup$ Feb 13 '17 at 17:44