  Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and
Meulenberg. In this paper, we introduce the notion of $q$-analogs of divisible
design graphs and show that all $q$-analogs of divisible design graphs come
from spreads, and are actually $q$-analogs of strongly regular graphs. Deza
graphs were introduced by Erickson, Fernando, Haemers and Hardy in 1999. In
this paper, we introduce $q$-analogs of Deza graphs. Further, we determine
possible parameters, give examples of $q$-analogs of Deza graphs and
characterize all non-strongly regular $q$-analogs of Deza graphs with the
smallest parameters.
