I am trying to interpolate monotone data with known data values and also known first derivative values at knots. If I used these values with cubic Hermite spline interpolation, Can I guarantee the monotonicity of the interpolation segments?
No, straightforward Hermite cubic interpolation does not guarantee monotonicity. However, there are other piecewise polynomial interpolation methods that do preserve monotonicity. Look up terms like "monotone spline" or "monotone interpolation". Here is one reference, which describes the Fritsch-Carlson method.