Is it true that Christian English monks adopted Norse hairstyles before the start of the Viking age?

I believe your question, or at least the documentary you are referencing, may be referring to a much later event. The 'complaint' you mention is found in a letter from Ælfric of Eynsham to an individual identified as 'brother Edward', and details three different behaviors the Abbot finds distressing; eating blood, assuming Danish customs, and eating on the toilet. Seems like a perfect fit to your question, except this takes place a couple hundred years later, with the letter being dated to about 1005-1010 CE.

From Wikipedia:

1005 is the other certain date we have for Ælfric, when he left Cerne for nobleman Æthelmær's new monastery in Eynsham, a long eighty-five-mile journey inland in the direction of Oxford. Here he lived out his life as Eynsham's first abbot, from 1005 until his death. After his elevation, he wrote his Letter to the Monks of Eynsham, an abridgment for his own monks of Æthelwold's De consuetudine monachorum, adapted to their rudimentary ideas of monastic life;

An excellent analysis of the Letter to Brother Edward, can be found in a transcript of an article which appeared in the Old English Newsletter. From the section discussing the haircut (emphasis mine):

The second section of the Letter deals with English men adopting Danish customs. Ælfric considers this shameful, and says that, according to books, anyone who practises the customs of heathen people and dishonors his own kindred by doing so will be cursed. He object in particular to those who tysliað eow on Denisc, ableredum hneccan and ablendum eagum. Tyslian is a rare word which means 'to dress'; ablered is otherwise unattested, but is connected to the word blere 'bald' by Bosworth and Toller and the Dictionary of Old English, and apparently means 'bare of hair'. The passage refers to Englishmen adopting a Danish style of haircut.

Note that though the letter is from the Abbot to a 'brother edward', at least this article's interpretation of the meaning of the section of the letter is referring to the behavior 'of Englishmen', not directly referencing the monks themselves.

The Alcuin letter is also discussed here as a reference Aelfric may have been familiar with. Another section of this article goes on to expand upon what exact hairstyle may have been meant, and brings up Saxon styles which was queried about in comments:

Neither Alcuin, the papal legates, not Wulfstan's Canon specifies which particular heathen fashion is indicated, but Ælfric's phrase ableredum hneccan and ablendum eagum 'with bared necks and blinded eyes', is very specific. It suggest a long fringe or bangs in front, and very short or shaved hair exposing the neck in back. This was very different from the way Anglo-Saxon men normally wore their hair. According to Gale Owen-Crocker's authoritative survey of Anglo-Saxon dress, Anglo-Saxon men of the tenth and eleventh centuries would usually have worn their hair short, and were commonly clean-shaven or had closely cropped beards, though kings are generally shown with moustaches and full beards. [29]

Even though we have no visual reference for the time, the Norman invasion of England is just a few years away. The Bayeux Tapestry is often referenced for its depiction of the Normans, particularly interesting is the hairstyle:

Note the bare neck, and the bangs depicted in front of the eyes. (The above OEN article also explains the connections between the Normans and the Danes, if you are unclear on that connection.)

So , a complaint by Aeflric to Brother Edward concerning Danish hairstyles certainly exists, but the timing is much later then the attack on Lindisfarne, and it probably wasn't referring to behavior of the monks themselves. This is still likely, in my opinion, to be the source of the information you recall.