Reactions which confirms the presence of cyclic structure of glucose
Yes, such a reaction exists. In fact, it can not only confirm the presence of ring forms in solution but also how prevalent they are.

In 1962, Walker Jr, Gee and McCready published Complete Methylation of Reducing Carbohydrates, in which they subject solutions of sugars in dry DMF to silver(I) oxide  and methyl iodide. The reaction proceeds according to scheme 1 below.


Scheme 1: Methylation of glucose with $\ce{Ag2O}$ and $\ce{MeI}$ as published by Walker Jr, Gee and McCready.

Before reporting on the products, they note:


  It was somewhat unexpected that methylation of
  aldoses would proceed so smoothly in the presence
  of silver oxide. Glucose, shaken with silver oxide in
  DMF at room temperature, is partially decomposed
  in several hours. 


Usually, silver(I) salts in solution are used in the Tollens test in which reducing sugars such as glucose reduce the silver ions to give elemental silver which forms a silver film on the inside of the glass. The presence of methyl iodide completely suppresses this reaction as they state in the next sentence:


  When methyl iodide is present
  in the reaction mixture, however, no silver mirror
  (indicating oxidation-reduction) is formed


As for the two step process that I have indicated, the authors state:


  In this reaction methylation
  of hydroxyl of the hemi-acetal group of glucose is
  essentially complete in one hour (as shown
  by loss of reducing function), but complete
  etherification of the remaining hydroxyls requires
  considerably longer.


With the experimental section detailing that the reactions were run ‘at least $\pu{16h}$’ which permits drawing the mechanism as a two-stage process as 1 can be isolated if the reaction is cut short.

Mechanistically, this reaction process is different from an acid-catalysed acetalisation with methanol. In the latter, an entire $\ce{OMe}$ group is attached to the sugar and the reactive species is an aldehyde form. This could be the open-chain aldehyde form (requires an additional reaction step) but most likely occurs via a cyclic aldehyde as depicted below. When using $\ce{Ag2O/MeI}$, the acetal oxygen is supplied by glucose as methyl iodide does not have any available oxygen. The two competing mechanisms are shown in scheme 2.


Scheme 2: Mechanisms of the acetalisation of glucose. (a) $\ce{H+/MeOH}$ and cyclic aldehyde intermediate; (b) $\ce{H+/MeOH}$ and chain-form aldehyde intermediate; (c) $\ce{Ag2O/MeI}$ single-step mechanism.

As these mechanisms show, acid-catalysed acetalisation need not start from the cyclic form to get acetal 1; it is possible to draw the reaction mechanism as originating from the chain form of glucose. But methylation with $\ce{MeI}$ requires the oxygen to be in place as an $\ce{OH}$ function that is deprotonated and attacks $\ce{MeI}$ which is not possible if the chain form were present in solution.

As to why the $\ce{C{1}}$ oxygen is methylated first, and the explanation is the higher acidity of an acetal hydroxy function when compared to an alcoholic one as the authors state:


  This difference in rate is expected
  from the difference in acidity (and ease of
  anion formation) of the hydrogens on the two different
  types of hydroxyl groups.


Furthermore, it bears stating that this reaction not only proves cyclic forms are present but also that they are prevalent. Methylation of the $\ce{C{1}}$ oxygen is rapid compared to all the other hydroxy groups meaning that they should be present in at least appreaciable ratios. Furthermore as mentioned above, under these conditions the reduction of silver(I) to a silver mirror – which requires the open-chain aldehyde form mechanistically – is practically entirely suppressed. Given that the Tollens test is a standardised method to detect glucose, the complete absence of all traces of silver film serves as a good indicator of the low concentration of open-chain aldehyde in solution.

The original paper is: H. G. Walker Jr, M. Gee, R. M. McCready, J. Org. Chem. 1962, 27, 1000–1002. DOI: 10.1021/jo01053a047.