How to predict amine donor in biochemical reactions (whether glutamine or aspartate)?
There are several very general concepts that you have to know about in order to see a pattern in biochemical reactions driven by enzymes. A quick answer is given at the very end. Amongst many concepts to understand, two main concepts are as following:
Enzyme are classified with a specific enzyme commission (EC) number, which is a type of ID number based on catalytic function. The first number groups the enzyme into an overall group. In example, enzymes that can transfer functional groups are placed in EC.2. Within EC.2 you can then subdivide into more specific functions, e.g. transaminases that transfer nitrogen containing groups have the the EC number of 2.6, and within this group again you can find enzyme that specifically transfer amine groups. Those enzymes will then have the EC number of 2.6.1 (aminotransferases), and you can then further specify into families, e.g. aspartate aminotransferases EC.2.6.1.1.
Enzymes folds within specific functional families is evolutionary conserved and highly specific. The three-dimensional shape of enzymes within a specific family, e.g. aspartate transaminase; EC 2.6.1.1, is very conserved (the amino acids sequence identity will also be very high)
and enzymes that are grouped into the same family will have highly similar overall structure and thus very similar mechanisms of catalysing reactions. In contrast, enzymes in completely different families, with different foldes and different amino-acid sequence will have completely different catalytic mechanisms.
To predict how some enzyme will catalyse a reaction, you need to find what enzyme it is related to. Here it will also be important to remember that structure is more conserved than sequence. A good starting point is to do a BLAST search with the protein sequence that you are investigating, note the you could also do a blast search against the PDB database to find related published protein structures.
Continuing with the exsample of amino-transferases, you can even subdivide the enzymes into where the catalytic reaction will take place. In example, you have alpha-aminotransferases or omega-aminotransferases, depending on the location on the substrate that the amino will we translocated to and from.
To answer your question:
Now that we covered some very general concepts about enzymes, it is important to know that their structure and sequence are specifically designed to accommodate a particular substrate. In other words, some enzymes will take aspartate as substrate and donate its amine group - while other type of enzymes will take glutamine as substate - and donate its amine group.

As a side-note. In proteins, the amine-group in aspartate would be part of the protein backbone (amide bond / peptide bond), and the side-chain (no amine group) would therefore be the only accessible part of this amino acid. It is therefore a good idea to specify if amino acids are substrates or catalytic residues (i.e. part of the enzyme) when asking about biochemical reactions.