The first search for the doubly heavy $ {{{{\varOmega}_{bc}^{0}}}} $ baryon and a search for the $ {{{{\varOmega}_{bc}^{0}}}} $ baryon are performed using $ {{{{\varOmega}_{bc}^{0}}}} $ collision data collected via the $ {{{{\varOmega}_{bc}^{0}}}} $ experiment from 2016 to 2018 at a centre-of-mass energy of $ {{{{\varOmega}_{bc}^{0}}}} $ , corresponding to an integrated luminosity of 5.2 $ {{{{\varOmega}_{bc}^{0}}}} $ . The baryons are reconstructed via their decays to $ {{{{\varOmega}_{bc}^{0}}}} $ and $ {{{{\varOmega}_{bc}^{0}}}} $ . No significant excess is found for invariant masses between 6700 and 7300 $ {{{{\varOmega}_{bc}^{0}}}} $ , in a rapidity range from 2.0 to 4.5 and a transverse momentum range from 2 to 20 $ {{{{\varOmega}_{bc}^{0}}}} $ . Upper limits are set on the ratio of the $ {{{{\varOmega}_{bc}^{0}}}} $ and $ {{{{\varOmega}_{bc}^{0}}}} $ production cross-section times the branching fraction to $ {{{{\varOmega}_{bc}^{0}}}} $ ( $ {{{{\varOmega}_{bc}^{0}}}} $ ) relative to that of the $ {{{{\varOmega}_{bc}^{0}}}} $ ( $ {{{{\varOmega}_{bc}^{0}}}} $ ) baryon, for different lifetime hypotheses, at 95% confidence level. The upper limits range from $ {{{{\varOmega}_{bc}^{0}}}} $ to $ {{{{\varOmega}_{bc}^{0}}}} $ for the $ {{{{\varOmega}_{bc}^{0}}}} $ ( $ {{{{\varOmega}_{bc}^{0}}}} $ ) decay, and from $ {{{{\varOmega}_{bc}^{0}}}} $ to $ {{{{\varOmega}_{bc}^{0}}}} $ for the $ {{{{\varOmega}_{bc}^{0}}}} $ ( $ {{{{\varOmega}_{bc}^{0}}}} $ ) decay, depending on the considered mass and lifetime of the $ {{{{\varOmega}_{bc}^{0}}}} $ ( $ {{{{\varOmega}_{bc}^{0}}}} $ ) baryon.