In The World of Ice and Fire, it is mentioned that:
Two estrangements are recorded but they did not last more than a year or two before the pair resumed their customary friendship. The Second quarrel is of note, as it was due to Jaehaerys' decision of 92 A.C. to pass over his granddaughter Rhaenys - the daughter of his deceased eldest son and heir, Prince Aemon - in favor of bestowing Dragonstone and place of heir apparent to his second eldest son, Baelon the Brave.
Alysanne saw no reason why a man should be preferred over a woman. And if Jaehaerys thought women of less use, then he would have no need of her.
The second quarrel continued for two years and was ended in 94 AC with intervention of their daughter Septa Maegelle (Formerly styled as Princess Maegelle Targaryen).
However, I could find no mention of the first quarrel.
My Question(s) are:
- What was it about and how long did it last?
- What made them reconcile after the first quarrel?
The following may help you in finding the answer.
- It was before 92 AC (duh!).
- It must have been mentioned somewhere in The World of Ice and Fire but apparently I can't find it. After searching a lot now, it appears that it may have never been mentioned (yet) but I am not sure.
- The first quarrel must have spanned over a period of no more than two years1
- It may have been about the decision to give away Prince Vaegon to the citadel for training as a maester at a young age. This is my speculation.
- It may have been about the decision to give away Princess Maegelle to the Faith. This is also my speculation.
- It may have been about death of their daughter Princess Viserra in a drunken horse race. Again, my speculation.
- It may have been about the decision to give Princess Saera to the faith against her will. The Princess escaped from Westeros and went to Essos, never to return. This is my speculation.
- Also, As suggested by S.Fruggiero, It is also plausible that first quarrel may have erupted when Wildlings attacked the Wall. Jaehaerys flew to aid of Night's Watch on his dragon along with Alysanne where they defeated the wildling host. It is possible that Jaehaerys may have wanted Alysanne to stay behind in safety of Red Keep while the Queen wanted to take a part in defense of the realm.
Reasoning for speculations
You may ignore the following if you don't wish to consider the possible leads I provided.
It is unclear how much role did Alysanne play in sending away her children to institutes like the Citadel or the Faith so it is plausible she may have resented such decisions, especially as they were made when the children were very young and in Saera's case, against her will.
Alysanne held great sway in court and often influenced decisions of the state but it is unclear if these decisions were made with her consent. Jaehaerys is known to be a wise King and he must have known what one of his descendants, King Daeron II said centuries later:
Too many dragons are just as dangerous as too few.
So for the good of the realm, Jaehaerys would be expected to dispatch his spare children (As line of succession was safe in lives of his eldest sons; Princes Aemon and Baelon) to orders like Citadel or the Faith, putting an end to their claims and minimizing the possibility of a war of succession. In such decisions, Jaehaerys did not even listen to Alysanne, such as his decision to passover their grand-daughter Rhaenys as he might have suspected that Westeros was not ready for a female monarch (Which proved to be true decades later when Dance of the Dragons started between two factions of House Targaryen).
So it is very much possible that Jaehaerys may have made those decisions unilaterally, without taking Alysanne on board, therefore causing the first Quarrel.
1. Alysanne and Jaehaerys are said to have married in 50 AC and they remained married for 46 years. Which makes one think that Alysanne died in 96 AC, ending their marriage. That is however incorrect as Alysanne died in either 100 AC or 99 AC as it is stated that Alysanne died within a year of suicide of Princess Gael, which happened in 99 AC. The 46-years marriage only stands true if we minus the years of their estrangement. As we know for sure, second quarrel last for two years, that brings us to 98 AC. If She died in 99 AC, the first quarrel must have lasted a year or less. Otherwise If She died in 100 AC, it must have lasted for two. See the Citadel thread on this topic. Also see the AWOIAF page.