According to Wikipedia,
Cross-conjugation is a special type of conjugation in a molecule, when in a set of three pi bonds only two pi bonds interact with each other by conjugation, while the third one is excluded from interaction. Whereas a normal conjugated system such as a polyene typically has alternating single and double bonds along consecutive atoms, a cross-conjugated system has an alkene unit bonded to one of the middle atoms of another conjugated chain through a single bond. In classical terms, one of the double-bonds branches off rather than continuing consecutively: the main chain is conjugated, and part of that same main chain is conjugated with the side group, but all parts are not conjugated together as strongly.
So there is no cross conjugation in the cation formed by removal of Bromine from P .
One further statement made wrong by the friend is that the immediate carbocation matters. This answer, though about benzene, clears the misconception that hybridisation is in any way dependent on time.
In terms of resonance structures, both the compounds are eqivalent
According to Wikipedia,
A field effect is the polarization of a molecule through space. The effect is a result of an electric field produced by charge localization in a molecule. This field, which is substituent and conformation dependent, can influence structure and reactivity by manipulating the location of electron density in bonds and/or the overall molecule. Field effects are relatively weak, and diminish rapidly with distance, but have still been found to alter molecular properties such as acidity
As there is no localisation of charge in this case, field effect is non-operational (localisation of charge is absence of resonance structures with charge present on different atoms in each).
Coming to the question, as both molecules would be equally stable in carbocation state, the only thing that can be done is to compare the relative stability of the original compounds: the one lesser stable in neutral state is more likely to undergo the reaction.
It is important to know the Inductive effect and $sp_3$ and $sp_2$ hybridisation to compare the molecules.
In P, the carbon atom bonded to benzene has two $sp_2$ hybridised carbons (with 33% s character) and the a bromine molecule adjacent, all three of which are electron-withdrawing, making the carbon more electron-deficient and thus, making the molecule more unstable than Q, which has only one $sp_2$ carbon adjacent to it. Thus, P will be more reactive
*By Jmancus0 - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=67323330