In anticipation of the upcoming Euclid Wide and Deep Surveys, we present optical emission-line predictions at intermediate redshifts from 0.4 to 2.5. Our approach combines a mock light cone from the Gaea semi-analytic model with advanced photoionisation models to construct emission-line catalogues. This allows us to self-consistently model nebular emission from H ii regions around young stars, and, for the first time with a semi-analytic model, narrow-line regions of active galactic nuclei (AGN) and evolved stellar populations. Gaea Mpc $, marks the largest volume this set of models has been applied to. We validate our methodology against observational and theoretical data at low redshift. Our analysis focuses on seven optical emission lines: Halpha , Hbeta S ii N ii O i O iii 5007$, and O ii 3727, 3729$. In assessing selection bias, we find that it will predominantly observe line-emitting galaxies, which are massive (stellar mass $ solarmass $), star-forming (specific star-formation rate $> 10^ yr^ $), and metal-rich (oxygen-to-hydrogen abundance $ logten(O/H)+12 > 8$). We provide percentages of emission-line populations in our underlying Gaea sample with a mass resolution limit of $10^ solarmass $ and an $H$-band magnitude cut of 25. We compare results with and without an estimate of interstellar dust attenuation, which we model using a Calzetti law with a mass-dependent scaling. According to this estimate, the presence of dust may decrease observable percentages by a further 20-30<!PCT!> with respect to the overall population, which presents challenges for detecting intrinsically fainter lines. We predict to observe around 30--70<!PCT!> of Halpha - N ii S ii -, and O iii -emitting galaxies at redshift below 1. At higher redshift, these percentages decrease below 10<!PCT!>. Hbeta O ii and O i emission are expected to appear relatively faint, thus limiting observability to at most 5<!PCT!> at the lower end of their detectable redshift range, and below 1<!PCT!> at the higher end. This is the case both for these lines individually and in combination with other lines. For galaxies with line emission above the flux threshold in the Euclid Deep Survey, we find that BPT diagrams can effectively distinguish between different galaxy types up to around redshift 1.8, attributed to the bias toward metal-rich systems. Moreover, we show that the relationships of Halpha and O iii +Hbeta to the star-formation rate, as well as the O iii -AGN luminosity relation, exhibit minimal, if any, changes with increasing redshift when compared to local calibrations. Based on the line ratios $ N ii /H N ii O ii $, and $ N ii S ii $, we further propose novel redshift-invariant tracers for the black hole accretion rate-to-star formation rate ratio. Lastly, we find that commonly used metallicity estimators display gradual shifts in normalisations with increasing redshift, while maintaining the overall shape of local calibrations. This is in tentative agreement with recent JWST data.
