Apparently very close. She is said to be her favourite daughter-in-law out of the 4 she's had do far.
From something I read here on TRF my understanding is the Queen was very solicitous of Sophie in the early years of her marriage when Sophie was having harrowing medical events happen around pregnancies.
For example, having Sophie ride with her (the Queen) at events so Sophie didn't have to walk or stress herself, etc. At that time they became close. (The fact that Edward and Sophie are the royal couple that always go abroad to attend continental royal events suggests that they are able to socialize effortlessly and agreeably).
It helps that Edward and Sophie live very close to Windsor. I have always had the sense that Edward and Sophie are the family that show up at Windsor to visit the Queen on a Sunday afternoon on a regular basis.
Others can correct me if I'm wrong.
Dare I dip into a comparison?
In many ways Sophie and Meghan share a similar background: both are coming from careers that require personability and an ability to not only get along with most anyone but be able to put people at their ease. I would think that alone was refreshing for the Queen.
It's why I think if Meghan and Harry get a house near to the Queen (and Charles), like in the Cotswolds, it is more likely that there will be a steady interaction between Harry's family and the Queen and Charles, as well as Edward and Sophie, and Anne's family, who all live in that area.
Sophie always speaks about the Queen with great admiration.
But everyone always speaks highly of the Queen.
It appears to be a given. To do otherwise would be unthinkable methinks. That's not to say Sophie does not have genuine admiration for the Queen. Especially given that they appear to be friends as well as in-laws.
BTW I have also read here on TRF that the Queen has been especially solicitous to Sophie's widowed father. Once his wife passed, the Queen made every effort to include the father at royal events, in carriage rides with her (the Queen), etc. That speaks to a close family relationship amongst that particular family grouping, I would say. Looks it to me.
LATER: Ah, this post....
I seem to recall Sophie saying once in an interview that when the Queen spends weekends at Windsor she always goes there for tea and takes her children with her.
Yes, this speaks to that kind of regular, every-day relationship. I'm not aware of any other family (outside of perhaps Anne's) that socialize with the Queen the way Edward and Sophie do. (Not even William and Kate).