Why did non-seasonally adjusted gdp usually drops on Q4 and Q1?

Output usually falls during certain quarters/months and for many countries it is in the winter. Tourists usually travel for vacations in summer when kids also get long summer holiday. Farm work is highly seasonal and there is more of it in summer than winter (as pointed out by @dismalscience in an excellent comment construction is seasonal too) etc. As a result GDP will fluctuate seasonally and this is in fact the reason why we seasonally adjust series for many research purposes. Although usually these fluctuations are not big enough to cause recessions.

Depending on geography and in which industries country specializes it will be affected differently by seasonal variation in output. There is no a priori reason why it should happen in Q1 and Q4 but for most countries it will be either during winter or summer. Since you do not mention which country you talk about there is not much what can be said about that.

If real GDP dropped in two consecutive quarters then usually it would qualify as a recession (for example that is definition given by Blanchard et al. Macroeconomics in glossary on pp 569), but that is not the only definition used in literature - there simply is no single agreed upon definition by all. Research on calling recessions often uses seasonally adjusted real GDP in order to identify recessions (e.g. Hamilton, 2011), but most definitions do not specify whether it should be seasonally adjusted or not.