This is so subtle I hope you can see it! Tweak at will.
Source Alexandre Naud
Light gray pattern with an almost wall tile-like appearance.
Source Markus Tinner
A dark gray, sandy pattern with small light dots, and some angled strokes.
Source Atle Mo
Not a pattern for fabrics, but one produced from a jpg of a stack of fabric items that was posted on Pixabay. The tile that this is based on can be had by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
One more updated pattern. Not really carbon fiber, but it’s the most popular pattern, so I’ll give you an extra choice.
Source Atle Mo
Number five from the same submitter, makes my job easy.
Source Dima Shiper
ZeroCC tileable stone texture, edited from pixabay. CC0
Source Sojan Janso
From a drawing in 'La Principauté de Liège et les Pays-Bas au XVIe siècle', Société des Bibliophiles Liégeois ,1887.
Source Firkin
Brushed aluminum, in a bright gray version. Lovely 2X as well.
Source Andre Schouten
Almost like little fish shells, or dragon skin.
Source Graphiste
If you’re sick of the fancy 3D, grunge and noisy patterns, take a look at this flat 2D brick wall.
Source Listvetra
A free green background pattern with a pattern of rhombuses on a seamless texture. Feel free to use it as a tiled background image on your web site.
Source V. Hartikainen
Colorful Floral Background 3 No Black
Source GDJ
Real snow that tiles, not easy. This is not perfect, but an attempt.
Source Atle Mo
From a drawing in 'Worsborough; its historical associations and rural attractions', Joseph Wilkinson, 1879.
Source Firkin
A gray background pattern with a texture of textile. Suits perfectly for web design.
Source V. Hartikainen
Wild Oliva or Oliva Wilde? Darker than the others, sort of a medium dark pattern.
Source Badhon Ebrahim