From this slide on page 25, the system of equations:
$$a = b \triangleleft (a \triangleleft b)$$ $$b = (a \triangleleft b) \triangleleft (b \triangleleft (a \triangleleft b))$$
Reduces down to:
$$(q^2 - q + 1)a \equiv (q^2 - q + 1)b \mod{n}$$
Using the example the presentation gives of $\mathbb{Z}_{7,3}$ we get that $(q^2 - q + 1)$ becomes $7$ so:
$$7a = 7b \mod{7}$$
which is trivially true for any $a$, $b$ in $\mathbb{Z}_n$. I am still unsure precisely what it means to be $Z_{n,q}$ (quandle) colorable. Does this mean that given a system of equations mod $n$ that for a particular choice of $q$ any choice of $a, b, c \ldots \in \mathbb{Z}_n$ will result in the modular congruence being satisfied?
To reiterate: in the example above we fixed $n$ to be $7$ and $q$ to be $3$ which allowed any choice of $a,b \in \mathbb{Z}_7$ to satisfy the system (in this case just one) of congruence relations.
So, let me try to answer the two questions I see.
First, a strand is an section of the diagram, which occurs between two under crossings, with no undercrossing between them. I.e. the part that you draw without picking up your pencil. Some people call these arcs, just fyi.
Second, a knot is said to be colored by a quandle if each strand (or arc) can be assigned an element of the quandle in a way that respects the quandle operation at the crossing.
In the pictures on page 25, each crossing has three strands. The strand that goes over the crossing, and the two strands that we see from the part that goes under.
As for $\mathbb{Z}_{7,3}$, the author is saying that to find a quandle that colors the trefoil, we need $n$ and $q$ which satisfy the equations. It happens that 7 and 3 work.
Hope this helps.