Not so much an answer but to elaborate on the issue..using EvaluationMonitor
we see it does quite a lot of evaluating outside the integration limits.
k = 9.011
\[Gamma] = .49
integrand = \[Eta]^2 Sqrt[\[Eta]^2 - \[Gamma]^2] BesselJ[0,
k \[Eta]]/((2 \[Eta]^2 - 1)^2 -
4 \[Eta]^2 Sqrt[(\[Eta]^2 - \[Gamma]^2) (\[Eta]^2 - 1)]);
{result, evals} =
Reap[NIntegrate[y = integrand, {\[Eta], 2, Infinity},
EvaluationMonitor :> Sow[{\[Eta], y}]]];
Show[{ListPlot[evals[[1]]],
Plot[integrand + 1/2, {\[Eta], 0, 10}, PlotStyle -> Red,
PlotPoints -> 1000, PlotRange -> {-20, 20}]}]

you can see it also stopped at 10. This has come up before (I cant find it though) but the speculation is that some internal change of variable has been made, and the change is not correctly reflected in the error message, or in what is passed to EvaluationMonitor
A manual change of variable fixes that issue (It still fails to converge as highly oscillatory however):
k = 9.011
\[Gamma] = .49
integrand = \[Eta]^2 Sqrt[\[Eta]^2 - \[Gamma]^2] BesselJ[0,
k \[Eta]]/((2 \[Eta]^2 - 1)^2 -
4 \[Eta]^2 Sqrt[(\[Eta]^2 - \[Gamma]^2) (\[Eta]^2 -
1)]) /. \[Eta] -> \[Eta] + 2 ;
{result, evals} =
Reap[NIntegrate[y = integrand, {\[Eta], 0, Infinity},
EvaluationMonitor :> Sow[{\[Eta], y}]]];
result
Show[{ListPlot[evals[[1]], PlotRange -> {{0, 10}, {-5, 5}}],
Plot[integrand, {\[Eta], 0, 10}, PlotStyle -> Red,
PlotRange -> {-5, 5}]}]

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
$\endgroup$NIntegrate
is probably using complex methods internally. The numbers in such error messages are typically garbage so don't worry about it. I think it diverges but if you need to prove it diverges that may be a difficult task. $\endgroup$