funcL = Integrate[
Simplify[PDF[BetaDistribution[a, b], t], 1 > t > 0], {t, 0, p},
Assumptions -> {0 < p < 1, b > 0, a > 0}];
funcU = Integrate[
Simplify[PDF[BetaDistribution[a, b], t], 1 > t > 0], {t, p, 1},
Assumptions -> {0 < p < 1, b > 0, a > 0}];
a = 2;(*I assume*)
b = 3;(*I assume*)
LowerP = FindRoot[funcL == 0.05, {p, 1/2}]
(* {p -> 0.0976115} *)
UpperQ = FindRoot[funcU == 0.05, {p, 1/2}]
(* {p -> 0.751395} *)
The results agree with the site
With a good hint of user Claude Mante
Quantile[BetaDistribution[a, b], #] & /@ {0.05, 0.95}
(* {0.0976115, 0.751395}*)