According to a paper by Walker et al. (2013) (found by googling "silverfish + silk", so not very hard to come by), silverfish can indeed produce silk. The paper also mentions that silverfish (Thysanura) and bristletails (Archaeognatha) use silk to transfer their sperm, and they also use it for "...tactile cue during courtship (Sturm, 1956)". This webpage (from NY State University) explains that male silverfish will deposit a spermatophore on a silk thread that is vertically suspended, and the spermatophore is later picked up by a female.
Silks are semi-crystalline solids in which protein chains are associated by intermolecular hydrogen bonding within ordered crystallites, and by entanglement within unordered regions. By varying the type of protein secondary structure within crystallites and the overall degree of molecular order within fibers, arthropods produce fibers with a variety of physical properties suited to many purposes. We characterized silk produced as a tactile stimulus during mating by the grey silverfish (Ctenolepisma longicaudata) using Fourier transform infrared spectroscopy, polarized Raman spectroscopy, gel electrophoresis and amino acid analysis. Fibers were proteinaceous-the main component being a 220 kDa protein-and were rich in Gln/Glu, Leu, and Lys. The protein structure present was predominantly random coil, with a lesser amount of beta-structure. Silk fibers could readily be solubilized in aqueous solutions of a mild chaotrope, sodium dodecyl sulfate, indicating protein chains were not cross-linked by disulfide or other covalent bonds. We conclude that entanglement is the major mechanism by which these silk proteins cohere into a solid material. We propose silks used as short-term tactile cues are subject to less stringent requirements for molecular order relative to other silks, allowing the random coil structure to be favored as an adaptation promoting maximal entanglement and adhesion.
The linked paper contains several references that should be useful if you want to look deeper into how silverfish use silk.
From what I've seen, they don't use silk for climbing. Silverfish can usually climb rough surfaces, but not smooth surfaces, which is why they are often found trapped in sinks, bathtubs etc.