Aims: We aim at detecting H$_2$O in the atmosphere of the hot Jupiter HD 209458 b and perform a multi-band study in the near infrared with CARMENES. Methods: The H$_2$O absorption lines from the planets atmosphere are Doppler-shifted due to the large change in its radial velocity during transit. This shift is of the order of tens of km s$^{-1}$, whilst the Earths telluric and the stellar lines can be considered quasi-static. We took advantage of this to remove the telluric and stellar lines using SYSREM, a principal component analysis algorithm. The residual spectra contain the signal from thousands of planetary molecular lines well below the noise level. We retrieve this information by cross-correlating the spectra with models of the atmospheric absorption. Results: We find evidence of H$_2$O in HD 209458 b with a signal-to-noise ratio (S/N) of 6.4. The signal is blueshifted by --5.2 $^{+2.6}_{-1.3}$ km s$^{-1}$, which, despite the error bars, is a firm indication of day-to-night winds at the terminator of this hot Jupiter. Additionally, we performed a multi-band study for the detection of H$_2$O individually from the three NIR bands covered by CARMENES. We detect H$_2$O from its 1.0 $mu$m band with a S/N of 5.8, and also find hints from the 1.15 $mu$m band, with a low S/N of 2.8. No clear planetary signal is found from the 1.4 $mu$m band. Conclusions: Our significant signal from the 1.0 $mu$m band in HD 209458 b represents the first detection of H$_2$O from this band, the bluest one to date. The unfavorable observational conditions might be the reason for the inconclusive detection from the stronger 1.15 and 1.4 $mu$m bands. H$_2$O is detected from the 1.0 $mu$m band in HD 209458 b, but hardly in HD 189733 b, which supports a stronger aerosol extinction in the latter.
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