Perhaps this is more what you look for.
For $\nu>-1$ and $x\in(1,\infty)$ we have the estimates $Q_\nu(x)\leq\min\left(q_\nu(x-1)^{-\nu-1},x^{-\nu-1}\left(q_\nu +\frac12\log\frac{x+1}{x-1}\right)\right)$ and $Q_\nu(x)\geq x^{-\nu-1}\max\left(q_\nu,\frac12\log\frac{x+1}{x-1}-\gamma-\psi(\nu+1) \right)$ where $q_\nu=\frac{\sqrt{\pi}\Gamma(\nu+1)}{2^{\nu+1}\Gamma(\nu+3/2)}$ and $\psi(x)=\frac{d}{dx}\log(\Gamma(x))$.
Also, in the case $\nu=1/2$ we have $q_{1/2}=\frac{\sqrt{2}\pi}{8}$.
To prove these estimates it is useful to:
(1) Use recurrence relations $\frac{d}{dx}\left((x^2-1)Q'_\nu(x)\right)=\nu(\nu+1)Q_\nu(x)$ $Q'_\nu(x)=\frac{\nu+1}{x^2-1}\left(Q_{\nu+1}(x)-xQ_\nu(x)\right)=\frac{\nu}{x^2-1}\left(xQ_\nu(x)-Q_{\nu-1}(x)\right) $
(2) Derive growth relations using (1). For example $\frac{d}{dx}\left(x^{\nu+1}Q_\nu(x)\right)=x^{\nu}Q'_{\nu+1}(x)\tag{ <0}$
(3) Have some convenient representations of $Q_\nu$ involving the leading term, e.g. $Q_\nu(x)=q_\nu x^{-\nu-1}\,_2F_1\left(\frac{\nu}{2}+1,\frac{1}{2}(\nu+1),\nu+\frac32;x^{-2}\right)$ (where $\,_2F_1$ is the hypergeometric function of Gauss).
For a details look up Proposition 3.4 in A Wiener Tauberian Theorem for Weighted Convolution Algebras of Zonal Functions on the Automorphism Group of the Unit Disc, Dahlner A. Contemp. Math no404, 2006.