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Since both sqrt(3)*x-y and -sqrt(3)*x+sqrt(3)-y must be positive to define your region, multiply the two together:

var('x y')
contour_plot((x^2) * cos(x*y), (x,0,1), (y,0,sqrt(3)/2),
             region=(sqrt(3)*x-y)*( -sqrt(3)*x+sqrt(3)-y), fill=True, contours=30)

Here's a live example.

Since both sqrt(3)*x-y and -sqrt(3)*x+sqrt(3)-y must be positive to define your region, multiply the two together:

var('x y')
contour_plot((x^2) * cos(x*y), (x,0,1), (y,0,sqrt(3)/2),
             region=(sqrt(3)*x-y)*( -sqrt(3)*x+sqrt(3)-y), region=(sqrt(3)*x-y)*(-sqrt(3)*x+sqrt(3)-y), fill=True, contours=30)

Here's a live example.