Valid term from quadratic sequence?
JavaScript (ES7), 70 bytes
Returns a Boolean value.

(a,b,c,t)=&gt;(t-=c,(a*=2)?(x=(b*b+2*a*t)**.5-b)%a&amp;&amp;(x+b+b)%a:b?t%b:t)==0

Try it online!
How?
For the sake of clarity, we define \$d = T_n-c\$. (The same variable \$t\$ is re-used to store this result in the JS code.)
Case \$a\neq0\$
The equation really is quadratic:
$$T_n=an^2+bn+c\\
an^2+bn-d=0$$
With \$a'=2a\$, the discriminant is:
$$\Delta=b^2+2a'd$$
and the roots are:
$$n_0=\frac{-b-\sqrt{\Delta}}{a'}\\
n_1=\frac{-b+\sqrt{\Delta}}{a'}$$
The equation admits an integer root if \$\sqrt{\Delta}\$ is an integer and either:
$$\begin{align}&amp;-b-\sqrt{\Delta}\equiv 0\pmod{a'}\\ \text{ or }&amp;-b+\sqrt{\Delta}\equiv 0\pmod{a'}\end{align}$$
Case \$a=0, b\neq0\$
The equation is linear:
$$T_n=bn+c\\
bn=d\\
n=\frac{d}{b}$$
It admits an integer root if \$d\equiv0\pmod b\$.
Case \$a=0, b=0\$
The equation is not depending on \$n\$ anymore:
$$T_n=c\\
d=0$$