I want to design some specific slope highpass filter to enhance speech; just like $1$ dB/octave, $2$ dB/octave and more. Is there a method to calculate the coefficients by specific slope?
The order of a filter determines its slope. Normally orders are positive integers (1, 2, 3, ...). So the slopes are fixed to multiples of approximately 3db/octave. However, taking the case of a Butterworth filter, someone has applied partial calculus to obtain fractional order Butterworth filters which will effectively allow you to chose the slope. If you do a Google search for fractional order Butterworth filters you will get back some papers on the subject. For instance here: https://arxiv.org/ftp/arxiv/papers/1210/1210.8194.pdf
If you need a different filter type you could apply the same thinking, but the maths may be more complicated for, say, an elliptic filter.