When Alicent arrives dramatically late to the wedding, wearing her own green gown instead of a Targaryen-themed red or black or gold dress, she sets herself firmly loyal to House Hightower and in opposition to what she sees as deceitful behavior from the Targaryens.
Her fallout with Rhaenyra is the result of a tragic series of misunderstandings, miscommunications, and misinterpretations of intentions. "House of the Dragon" is taking the time to show us how Alicent is isolated and traumatized by performing her "duty" to both her own family and the realm. She is scared, lonely, and increasingly resentful of Rhaenyra's seeming freedom to do as she pleases.
Alicent follows the Faith of the Seven, a religion that deems incest (between uncles and nieces, like Daemon and Rhaenyra) and sex outside of marriage as sinful. The Faith makes an exception for Targaryens, since their practice of intermarriage is born from a desire to preserve the dragonriding bloodlines of Valyria.
The themes of patriarchy and women's agency run strongly throughout "House of the Dragon" so far, and Alicent is emblematic of the internalized misogyny that some women feel towards people who are trying to dismantle the systems of oppression.
Alicent is made to resent Rhaenyra's presumed "freedom" from the duties the Alicent was forced into at a young age. And that resentment is only going to fester as the story continues.
For more on "House of the Dragon," read our analysis of the stag hunt scene and other details you might have missed this season.