How does Pat Martino's Minor Concept relate to his Diminished Concept? Are they completely unrelated or are melodic minor scales and diminished chords related in some way?
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There's no direct relation between the two concepts. The so-called diminished concept is just a way of memorizing the shapes of the four inversions of dominant seventh chords by lowering one of the four notes of the diminished chord.
By its symmetrical structure, the four inversions of the diminished chord have the same shape on the guitar, that's why they're easy to remember. This is not the case with other seventh chords (and other 4-part chords in general). The special thing about the dominant seventh chord is that it can be easily constructed from a diminished chord by lowering one note. E.g., from an Edim7 chord in a drop-2 shape on the top four strings
X X 2 3 2 3
we can easily construct the four dominant seventh chords
Eb7: X X 1 3 2 3
Gb7: X X 2 3 2 2
A7: X X 2 2 2 3
C7: X X 2 3 1 3
The minor concept is different because it is about improvising and how to think about scales. Martino basically reduces most chord scales to their relative (melodic) minor scale. So over an altered E7 chord he would think F minor (F melodic minor equals the E altered scale). Similarly, over a Dm7(b5) chord he would also think in F (melodic) minor, effectively using locrian ♮2 over the half-diminished chord.
Unlike the harmonic minor scale, the melodic minor scale is not very useful for playing over diminished chords. Note that no seventh chord built on any scale degree of the melodic minor scale is a diminished chord, whereas the harmonic minor scale has a diminished chord built on its seventh scale degree.