The groups have trivial center, as pointed out by Sam Nead. Another, more combinatorial, way to show this is to apply the algorithm from [1], which decides whether any given one-relator group has a non-trivial center or not. If one wishes to be very explicit, one can even compute an explicit generating set for a finite index surface group inside the groups, see [2], particularly §5, where the Reidemeister-Schreier method is used to find a one-relator presentation on $4q-2$ generators for such a finite index subgroup. 

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[1] <cite authors="Baumslag, G.; Taylor, Tekla">_Baumslag, G.; Taylor, Tekla_, [**The centre of groups with one defining relator**](http://dx.doi.org/10.1007/BF02063216), Math. Ann. 175, 315-319 (1968). [ZBL0157.34901](https://zbmath.org/?q=an:0157.34901).</cite>

[2] <cite authors="Baumslag, Gilbert; Troeger, Douglas">_Baumslag, Gilbert; Troeger, Douglas_, Virtually free-by-cyclic one-relator groups. I., Fine, Benjamin (ed.) et al., Aspects of infinite groups. A Festschrift in honor of Anthony Gaglione. Papers of the conference, Fairfield, USA, March 2007 in honour of Anthony Gaglione’s 60th birthday. Hackensack, NJ: World Scientific (ISBN 978-981-279-340-9/hbk). Algebra and Discrete Mathematics (Hackensack) 1, 9-25 (2008). [ZBL1188.20023](https://zbmath.org/?q=an:1188.20023).</cite>